Sum With Harmonic Numbers (math puzzle)

Let f(x) be any real->real periodic analytic (about 0) function,
where f(x) = f(x+y) for all x and a fixed y.
Also, f(x) = sum{k=0 to oo} a(k) x^k/k!,
and {a(k)} is such that the sum below (which you are to evaluate)
converges absolutely.

Let H(0,m) = 1/m, for m = any positive integer.
And let H(n,m) = sum{k=1 to m} H(n-1,k), for m and n= any positive
integers.
(So, H(n,m) is a generalization of the harmonic numbers, where H(1,m) =
H_m, the standard harmonic number.)
(Conway and Guy's "Book of Numbers" {I think is the title} talks about
this generalization of harmonic numbers. So do I write of these
harmonic numbers in countless posts to sci.math.)


Evaluate, in terms of n and f(x) and y (where y is the period of f):

oo
--- 2k
\ H(n,2k) a(2k) y
/ ----------------- = ?
--- (2k +2n -1)!
k=1

In linear mode:

sum{k=1 to oo} H(n,2k) a(2k) y^(2k) /(2k +2n-1)! = ?

I will give my solution in a few days if no one else gets it sooner.

thanks,
Leroy Quet

.

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